In Turbo equalization and soft interference cancellation, demodulation and decoding are done in an iterative way. In every iteration, a signal regeneration step computes the soft symbols (the expected value of the symbols), sn, and their variances, σn2, for every symbol Sn. The soft symbols can be used for interference cancellation, whereas their variances can be used in formulating linear equalization coefficients. These two quantities are defined as snE[Sn]=Σi=0M-1xi·Pr[Sn=xi],  (1a)σn2E[|Sn|2]−| sn|2=Σi=0M-1|xi|2·Pr[Sn=xi]−| sn|2,  (1b)where M is the size of the constellation used for transmission and xi is the ith member of this constellation. Note that the first term on the right-hand side of equation (1b) represents the second moment of symbol value. Thus, the soft symbol variance σn2, for every symbol Sn can be determined by the second moment of symbol value and the expected value of the symbols. The relationship between the variance, second moment, and expected value of a random variance is well known in the probability and random process literature.
The signal regeneration step requires the probability distribution of the symbol Sn, Pr[Sn=xi], i=0, 1, . . . , M−1, to calculate these quantities. What is actually available to the signal regeneration step after every turbo iteration are the bit log-likelihood ratios (LLR), rather than the symbol probability distributions. The bit LLR of the kth bit of the nth symbol, Bn,k, is
                              L                      n            ,            k                          ⁢                  =          Δ                ⁢                  log          ⁢                                                    Pr                ⁡                                  [                                                            B                                              n                        ,                        k                                                              =                    0                                    ]                                                            Pr                ⁡                                  [                                                            B                                              n                        ,                        k                                                              =                    1                                    ]                                                      .                                              (        2        )            Let the bit representation of the symbol xi in the constellation bexi[bi,0, . . . ,bi,Q-1],  (3)where bi,kε{0,1} and Q=log2 M. Define the bipolar bits, b′i,k1−2·bi,k, so that b′i,k=1 when bi,k=0 and b′i,k=−1 when bi,k=1. Then, it can be shown that the probability Pr[Sn=xi] is
                                          Pr            ⁡                          [                                                S                  n                                =                                  x                  i                                            ]                                ⁢                      =            Δ                    ⁢                                                    ∏                                  k                  =                  0                                                  Q                  -                  1                                            ⁢                                                          ⁢                              [                                                      B                                          n                      ,                      k                                                        =                                      b                                          i                      ,                      k                                                                      ]                                      =                                          C                n                            ·                              exp                ⁡                                  [                                                            1                      2                                        ⁢                                                                  ∑                                                  k                          =                          0                                                                          Q                          -                          1                                                                    ⁢                                                                        b                                                      i                            ,                            k                                                    ′                                                ·                                                  L                                                      n                            ,                            k                                                                                                                                ]                                                                    ,                            (        4        )            where Cn is a scaling factor independent of i or k but depends on n. That is, it is independent of the constellation but must be computed for every symbol Sn.
Existing solutions involve either high complexity or too much inaccuracy. For example, to get exact values of the expected value and variance with a general 16-QAM, the 16 probabilities Pr[Sn=xi] for i=0, 1, . . . , 15, must be computed from the bit LLRs according to (4), then the soft symbols in (1a) and finally the variances in (1b). Note that (1a) and (1b) cannot be done in parallel. First the means must be computed and then the variances. (The computation of (1a) and the first term on the right-hand side of (1b) however can be performed in parallel.) The mean (1a) requires sixteen multiplications and fifteen additions. The variance (1b), on the other hand, requires seventeen multiplications and sixteen additions.
On the other hand, a low complexity approach to compute the variances isσn2≈1−| sn|2.  (5)This approximation gives results whose quality deteriorates with the turbo iterations because E[|Sn|2] becomes significantly different from 1. The approximate variances might even be negative and require clipping.